DTIC ADA595761: Reservoir Model of Ebb-Tidal Shoal Evolution and Sand Bypassing

31 Pages·0.49 MB·English

by  Defense Technical Information Center

In: J. Waterway, Port, Coastal, and Ocean Engineering, 126(3): 305-313, 2000.

Reservoir Model of Ebb-Tidal Shoal Evolution and Sand Bypassing

By Nicholas C. Kraus,1 Member, ASCE

ABSTRACT
A mathematical model is presented for calculating the change in volume and sand-bypassing
rate at ebb-tidal shoals. Conceptually and mathematically, the ebb-tidal shoal is distinguished
from bypassing bars that emerge from it and from attachment bars where the bypassing bars
merge with the beach. The volumes and bypassing rates of these morphologic entities are
calculated by analogy to a reservoir system, where each reservoir can fill to a maximum
(equilibrium) volume. The ratio of the input longshore sand transport rate and the equilibrium
volume of the morphologic feature is found to be a key parameter governing morphologic
evolution. The analytical model gives explicit expressions for the time delays in evolution of the
bypassing bar and the attachment bar, which are directly related to the delays in sand bypassing.
Predictions of morphology change agree with observations made at Ocean City Inlet, Maryland.
Examples of extension of the model by numerical solution are given for a hypothetical case of
mining of an ebb-tidal shoal and for an idealized case of bi-directional longshore sand transport,
in which updrift and downdrift bypassing bars and attachment bars are generated.
Key Words: ebb-tidal shoal, longshore sand transport, sand bypassing tidal inlet, tidal prism.
INTRODUCTION
Inlet ebb-tidal shoals store sand transported by littoral and tidal currents, and they also
transfer or bypass sand to the down-drift beach according to the state of maturity of the shoal,
wave conditions, magnitude of the tidal prism, and other factors. Navigation projects and shore-
protection projects located in the vicinity of inlets must take into account the sand volume
storage and bypassing function of ebb-tidal shoals. For example, sand is often mechanically
bypassed at inlets, ebb-tidal shoals are a potential source of sand for the nourishment of local
beaches, and navigation channels must be maintained through ebb-shoals. Bruun and Gerritsen

1Res. Physical Scientist, U.S. Army Engr. Res. and Development Center, Coastal and Hydraulics
Lab., 3909 Halls Ferry Road, Vicksburg, MS 39180-6199. [email protected].
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Reservoir Model of Ebb-Tidal Shoal Evolution and Sand Bypassing
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Kraus, Nicholas C.
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13. SUPPLEMENTARY NOTES
14. ABSTRACT
A mathematical model is presented for calculating the change in volume and sand-bypassing rate at
ebb-tidal shoals. Conceptually and mathematically, the ebb-tidal shoal is distinguished from bypassing
bars that emerge from it and from attachment bars where the bypassing bars merge with the beach. The
volumes and bypassing rates of these morphologic entities are calculated by analogy to a reservoir system,
where each reservoir can fill to a maximum (equilibrium) volume. The ratio of the input longshore sand
transport rate and the equilibrium volume of the morphologic feature is found to be a key parameter
governing morphologic evolution. The analytical model gives explicit expressions for the time delays in
evolution of the bypassing bar and the attachment bar, which are directly related to the delays in sand
bypassing. Predictions of morphology change agree with observations made at Ocean City Inlet, Maryland.
Examples of extension of the model by numerical solution are given for a hypothetical case of mining of an
ebb-tidal shoal and for an idealized case of bi-directional longshore sand transport, in which updrift and
downdrift bypassing bars and attachment bars are generated.
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(1959, 1960) first described the bypassing function of inlets and classified the type of bypassing
and navigation difficulty to the ratio of spring tidal prism and the total amount of littoral material
transported to the inlet annually. FitzGerald (1988) reviewed natural bypassing mechanisms at
tidal inlets, and Gaudianao and Kana (2000) recently quantified episodic bypassing through
cycles of shoal attachment observed at inlets along the South Carolina coast.
Walton and Adams (1976), Marino and Mehta (1988), and Hicks and Hume (1996)
quantified the sand storage capacity of ebb-tidal shoals by relating their volume at or near
equilibrium to the tidal prism. Similarly, Gibeaut and Davis (1993) related areal extent of ebb-
tidal shoals to the tidal prism. Prior to those works, LeConte (1905), O’Brien (1931, 1969),
Johnson (1973), Jarrett (1976) and others showed that the inlet channel equilibrium cross-
sectional area was almost linearly related to the tidal prism. These types of empirical
observations of complex systems indicate that mathematical modeling of the main morphologic
features of inlets may yield other results of value to coastal engineering and science.
Mathematical modeling of large-scale morphology change at inlets is a relatively new area of
research and has been reviewed by De Vriend (1996a, 1996b). A fruitful approach appears to lie
in “aggregate” models, through which a system of large-scale geomorphic features, such as
shoals (also called “deltas” in the literature), channels, and beaches are represented by a small
number of attributes, typically their volume, area, or distance from a reference line. The
aggregate concept differs from a process-based approach that requires information be finely
resolved in time and space on such quantities as water depth; wave height, period, and direction;
bottom roughness; and sediment transport rate. Longer time scales, typically spanning months to
centuries, explicitly or implicitly accompany the aggregated feature.
Di Silvio (1989) introduced the concept of equilibrium sediment concentrations at locations
around an inlet and lagoon system. The resultant system of equations represents evolution or
adaptation of the morphological aggregates to external changes in the forcing condition, such as
slow rise in sea level. Although not expressed in terms of readily available coastal-engineering
information, time and space scales for large-scale behavior emerged from his model.
De Vriend, et al. (1994) schematized the outer area of the ebb-tidal shoal with a two-line
numerical model, in which the margin of the outer shoal was related to the inshore line
(shoreline), with both longshore and cross-shore sediment transport represented. Stive, et al.
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(1998) present an aggregate model describing the interactions among a lagoon or basin, ebb-tidal
shoal, and adjacent beaches. The model requires the assumption of equilibrium states for the
various morphological features and equilibrium sediment concentrations, similar to the Di Silvio
(1989) model. The Stive, et al. (1998) model calculates with input of process-based information
and is solved numerically.
Mehta, et al. (1996) developed a process-based model of ebb-shoal volume change, in which
calculation efficiency and robustness were obtained by requiring the shoal to have constant plan-
form area, leaving only its elevation or thickness to be calculated. Increase in volume of the
shoal was controlled by a parameter related to the ratio of wave power and tidal power (O’Brien
1971). Waves and tidal current drive the approach to equilibrium volume according to the
critical shear stress for sediment motion. The numerical model was run to simulate ebb-shoal
growth at several inlets in Florida, and observed trends were obtained. Episodic wave conditions
and ebb-shoal mining were also investigated. Devine and Mehta (1999) developed a similar
model for calculating cross-shore sediment transport and change in volume of an ebb-tidal shoal
as caused by storm surge and storm waves. Although describing phenomena at much shorter
time scales (hours to days) than associated with aggregate models, the approach of Devine and
Mehta (1999) shares similarity to the model presented here in resting on equilibrium
assumptions, in particular on an equilibrium profile for the seaward face of the shoal.
This paper introduces a mathematical aggregate model of volume change and sand bypassing
at inlet ebb shoals, based on a new approach, that of a reservoir analogy. Under simplifying
assumptions, a closed-form analytic solution is obtained in terms of commonly available or
inferable engineering parameters. Predictions of the model are compared with measurements
available for Ocean City Inlet, Maryland. Examples of extension of the model by numerical
solution are given for the hypothetical situation of ebb-shoal mining at Ocean City and for an
idealized case of bi-directional longshore sand transport.
CONCEPTUAL MODEL
A time-dependent model is sought that operates on the temporal and spatial scales associated
with the entire (aggregated) morphological form of an ebb-tidal shoal. Four assumptions are
initially invoked to arrive at the model:
1. Mass (sand volume) is conserved.
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2. Morphological forms and the sediment pathways among them can be identified, and the
morphologic forms evolve while preserving identity.
3. Stable equilibrium of the individual aggregate morphologic form9s) exists.
4. Changes in meso- and macro-morphological forms are reasonably smooth.
The above general rules evidently pertain to many types of aggregate models. It the present
context, a fifth is added that material composing an ebb-tidal shoal is predominately transported
to and from it through longshore transport. This latter assumption can be relaxed in extension of
the model.
Assumption 2 is required so that the morphologic form can be identified and tracked. It is a
basic assumption in aggregate modeling and distinguishes this type of model from particle-
based, micro-scale models. Particle-based models require a balance of complex, time-varying
physical forces to maintain the identity and form of the morphologic feature. Great
simplification results if the existence of the feature is assumed and quantified to some level,
which is justified for ebb-tidal shoals by many observations (e.g., Bruun and Gerritsen 1959;
Dean and Walton 1975; Walton and Adams 1976; Marino and Mehta 1988; Gibeaut and Davis
1993, Gaudiano and Kana 2000).
Concerning Assumption 3, stable equilibrium of a morphological form is an idealized state
that would be attained if all competing forces creating and molding the form were constant
through time. Equilibrium is considered stable if a feature returns to this state under perturbation
by small time-varying forces. Processes known to maintain ebb-tidal shoals are waves and
longshore sand transport, the ebb-tidal jet or prism, tidal range, and gravity (FitzGerald 1988).
Because the competing processes vary about a representative state, the volume and form of an
ebb-tidal shoal varies about the idealized static equilibrium. Implicit in Assumption 3 is that the
inlet morphologic system and adjacent beaches autonomously return toward an equilibrium state
after being perturbed by a relatively small force. However, under large impressed force or
action, such as significant mining of the ebb shoal, dredging of a deeper or wider channel,
placement or modification of jetties, or a disruptive storm, the inlet-beach system may move to a
new state compatible with the overall new conditions and constraints. Eventually, a new
equilibrium condition might be reached for this state. Assumption 4 is needed to take time
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derivatives, and it is physically justified in that macro-scale features possess enormous inertia
(Kraus 1998; Stive, et al. 1998).
In the present work, the ebb-shoal complex is defined as consisting of the ebb shoal proper,
one or two ebb-shoal bypassing bars (depending on the balance between left- and right-directed
longshore transport), and one or two attachment bars. These features are shown schematically in
Fig. 1 and the pattern of wave breaking on the crescentic ebb-shoal complex at Ocean City Inlet,
Maryland is shown in Fig. 2. The model distinguishes between ebb-tidal shoal proper (hereafter,
ebb shoal), typically located in the confine of the ebb-tidal jet, and the ebb-shoal bypassing bar
(hereafter, bypassing bar) that grows toward shore from the ebb shoal, principally by the
transport of sediment alongshore by wave action. The bar may shelter the leeward beach from
incident waves such that a salient might form – similar to the functioning of a detached
breakwater (Pope and Dean 1986), initiating creation of the attachment bar.
Previous authors have combined the ebb shoal and the bar(s) protruding from it into one
feature referred to as the ebb shoal. Here, the shoal and bypassing bars are distinguished because
of the different balance of processes. When a new inlet is formed, the shoal first becomes
apparent within the confine of the inlet ebb jet, and bypassing bars have not yet emerged.
Bypassing bars are formed by sediment transported off the ebb shoal through the action of
breaking waves and the wave-induced longshore current (tidal and wind-induced currents can
also play a role). A bar cannot form without an available sediment source, similar to the growth
of a spit (as modeled mathematically by Kraus 1999). In this sense, bypassing bars are
analogous to the spit platform concept of Meistrell (1972) in which a subaqueous sediment
platform develops from the sediment source prior to the visually observed subaerial spit.
Bypassing bars grow in the direction of predominant transport as do spits. At inlets with nearly
equal left- and right-directed longshore transport or with a small tidal prism, two bars can emerge
from the ebb shoal creating a nearly concentric halo about the inlet entrance. As the bypassing
bar merges with the shore, an attachment bar is created, thereby transporting sand to the beach.
At this point in evolution of the ebb-shoal complex, substantial bypassing of sand can occur from
the up-drift side of the inlet to the down-drift side.
In this conceptualization, if an inlet is created along a coast, the littoral drift is intercepted to
deposit sand first in the channel and ebb shoal. This material joins that volume initially jetted
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offshore when the barrier island or landmass was breached. Over time, a bar emerges from the
shoal and grows in the predominant direction of drift. After many years, as controlled by the
morphologic or aggregate scale of the particular inlet, an attachment bar may form on the down-
drift shore. At this stage, significant sand bypassing of the inlet can occur, re-establishing in
great part the transport down-drift that existed prior to formation of the inlet. The model
presented below can describe the evolution of an ebb-shoal complex from initial cutting of the
inlet, as well as changes in morphologic features and sand bypassing resulting from engineering
actions such as mining or from time-dependent changes in wave climate (for example, seasonal
shifts).
RESERVOIR AGGREGATE MODEL
The conceptual model of the ebb-shoal complex described in the preceding section is
represented mathematically by analogy to a reservoir system, as shown in Fig. 3. It is assumed
that sand is brought to the ebb shoal at a rate Q , and the volume V in the ebb shoal at any time
in E
increases while possibly “leaking” or bypassing some amount of sand to create a down-drift
bypassing bar. The input rate Q typically is the sum of left- and right-directed longshore sand
in
transport. For the analytic model presented, a predominant (uni-directional) rate is taken, but
this constraint is not necessary and is relaxed in a numerical example given below.
The volume V of sand in the shoal can increase until it reaches an equilibrium volume V
E Ee
(the subscript e denoting equilibrium) according to the hydrodynamic conditions such as given
by Walton and Adams (1976). As equilibrium is approached, most sand brought to the ebb shoal
is bypassed in the direction of predominant transport. Similarly, the bypassing bar volume V
B
grows as it is supplied with sediment by the littoral drift and the ebb shoal, with some of its
material leaking to (bypassing to) the attachment bar. As the bypassing bar approaches
equilibrium volume V , most sand supplied to it is passed to the attachment bar V . The
Be A
attachment bar transfers sand to the adjacent beaches. When it reaches its equilibrium volume
V , all sand supplied to it by the bar is bypassed to the down-drift beach. The model thus
Ae
requires values of the input and output rates of transport from each feature, and their respective
equilibrium volumes.
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Analytical Model
Simplified conditions are considered here to obtain a closed-form solution that reveals the
parameters controlling the aggregated morphologic ebb-shoal complex. In the absence of data
and for convenience in arriving at an analytical solution, a linear form of bypassing is assumed.
The amount of material bypassed from any of the morphological forms is assumed to vary in
direct proportion to the volume of the form (amount of material in a given reservoir) at the
particular time. Therefore, the rate of sand leaving or bypassing the ebb shoal, (Q ) , is
E out
specified as
V
(Q ) = E Q (1)
E out V in
Ee
in which Q is taken to be constant (average annual rate), although this is not necessary.
in
The continuity equation governing change in V can be expressed as
E
dV
E =Q −(Q ) (2)
dt in E out
where t = time. For the present situation with (1), it becomes
dV  V 
E =Q 1− E  (3)
dt in V
 
Ee
With the initial condition V (0) = 0, the solution of (3) is
E
( −αt)
V =V 1−e (4)
E Ee
in which
Q
α= in (5)
V
Ee
The parameter α defines a characteristic time scale for the ebb shoal. For example, if Q =
in
1×105 m3/year and V = 2×106 m3, which are representative values for a small inlet on a
Ee
moderate-wave coast, then 1/α = 20 years. The shoal is predicted to reach 50% and 95% of its
equilibrium volume after 14 and 60 years, respectively, under the constant imposed transport
rate. These timeframes are on the order of those associated with development of inlet ebb shoals.
7
The characteristic time scale given by α has a physical interpretation by analogy to the well-
known model of bar bypassing introduced by Bruun and Gerritsen (1960) and reviewed by
Bruun, et al. (1978). Bruun and Gerritsen (1960; see also, Bruun and Gerristsen 1959)
introduced the ratio r as
P
r = (6)
M
tot
in which P = tidal prism, and M = average annual littoral sediment brought to the inlet. Inlets
tot
with a value of r > 150 (approximate) tend to have stable, deep channels and are poor “bar
bypassers” from up drift to down drift, whereas inlets with r < 50 (approximate) tend toward
closure and are good bar bypassers. Because the equilibrium volume of the ebb-tidal shoal is
approximately linearly proportional to the tidal prism, V ∝ P (Walton and Adams 1976), α is
Ee
proportional to 1/r. The reservoir aggregate model therefore contains at its center a concept
widely accepted by engineers and geomorphologists. Established here through the continuity
equation, the reservoir model gives theoretical justification for the Bruun and Gerritsen ratio by
the appearance of α.
The volume of sediment (V ) that has bypassed the shoal from inception of the inlet to time
E out
t is the difference between the amount that arrived at the shoal and that remaining on the shoal:
(V ) =Q t−V (7)
E out in E
The rate of sand arriving at the bypassing bar (Q ) equals the rate of that leaving the shoal
B in
(Q ) = d(V ) /dt, or
E out E out
dV
(Q ) =(Q ) =Q − E
B in E out in dt
(8)
V
= E Q
V in
Ee
which recovers (1) by volume balance. The right side of (8) can be expressed as αV , again
E
showing the central role of the parameter α.
Continuing in this fashion, the reservoir aggregate model yields the following equations for
the volume of the bypassing bar,
8
V =V (1−e−βt′), β= Qin , t′=t− VE (9)
B Be V Q
Be in
and for the volume of the attachment bar,
Q V
V =V (1−e−γt′′), γ = in , t′′=t′− B (10)
A AE V Q
AE in
The quantities β and γ are analogous to α in representing time scales for the bypassing bar and
attachment bar, respectively.
The quantities t′ and t′′ in (9) and (10) can be interpreted as lag times that delay
development of the bar and attachment, respectively. To see this explicitly, Taylor expansions
for small relative time give t′≈αt2 /2 and t′′≈α2βt4 /8, as compared to growth of the ebb shoal
given by αt. The interpretation is that after creation of an inlet, a certain time is required for the
bypassing bar to receive a significant amount of sand from the shoal and a longer time for the
attachment bar or beach to receive sand. Similarly, modification of, say, the ebb shoal as
through sand mining will not be observed immediately in the bypassing at the beach because of
the time lags in the system.
A unique “crossover” time t occurs when the volume of material leaving the shoal equals the
c
volume retained, (V ) = V . After the crossover time, the shoal bypasses more sediment than it
E out E
retains, characterizing the time evolution of the ebb shoal and its bypassing functioning. The
crossover time is determined from (7) to be
1.59
t = (11)
c α
Finally, by analogy to (8), the following equations are obtained for the bypassing rate of the
bar (Q ) , which is equal to the input of the attachment (Q ) , and the bypassing rate of the
B out A in
attachment (Q ) , which is also the bypassing rate or input to the beach, (Q ) :
A out beach in
V V
(Q ) = E B Q =(Q ) (12)
B out V V in A in
Ee Be
V V V
(Q ) = E B A Q =(Q ) (13)
A out V V V in beach in
Ee Be Ae
9